Exceptional points (EPs) in non-Hermitian systems have attracted broad interest in recent years due to their topological properties and applications in sensing technology [1–4]. The energy eigenvalues of non-Hermitian systems coalesce at the EPs [5], and the complex eigenspectra possess the topology of complex Riemann surfaces. EPs exist in various physical systems, including microwave [6,7], photonic [8–27], optomechanical [28,29], atomic [30,31], electronic [32–35], condensed matter [36,37], acoustic [38], and other systems [39–41]. The nontrivial topological property of EPs makes the spectra of non-Hermitian systems fundamentally different from those of Hermitian systems [42–50]. Topological dynamics by adiabatic encircling of EPs has recently been realized in optomechanical experiments [28,51], which can be simulated by microwave and optical waveguide systems [52–57]. On the other hand, the nonlinear signature of the complex spectra around the EP singularities has been used for advanced sensing technology with EP-enhanced sensitivity [13,14,20,27,35,58]. So far, EP-enhanced sensing has mainly been focused on systems above the optical wavelength scale due to diffraction limits. Local surface plasmon-exciton hybrid systems can offer sensing devices beyond the diffraction limit and with intrinsic nanoscale spatial resolution [59–62]. It is intriguing to develop a scheme to implement the EPs in plasmon-exciton systems for enhanced sensitivity while maintaining the advantages that plasmon-exciton sensors already possess.

Here, we propose an experimental scheme to realize EPs in a plasmon-exciton hybrid system that consists of a gold nanorod (GNR) and monolayer ${\mathrm{WSe}}_2$. By tuning the geometric parameters of the hybrid system, we observe an EP in the complex eigenspectrum. We then apply the plasmon-exciton system operating near the EP to sense the variation of environmental refractive indices, and demonstrate significantly enhanced nanoscale sensitivity by numerical experiments.

Our plasmon-exciton hybrid system is depicted in Fig. 1(a). A monolayer of ${\mathrm{WSe}}_2$ is placed on a glass substrate and coated with a thin layer of ${\mathrm{HfO}}_2$. A GNR is then placed on top of ${\mathrm{HfO}}_2$. The GNR diameter is 30 nm, with variable lengths around 100 nm. The size of the monolayer ${\mathrm{WSe}}_2$ is set as $5\mathrm{\mu m}\times 10\mathrm{\mu m}$, so it is much larger than the size of the GNR. The interaction between the longitudinal plasmon mode of GNR and the exciton mode of ${\mathrm{WSe}}_2$ leads to two hybrid modes. We can describe the plasmon-exciton hybrid system with the coupled mode equation: $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=-i\left(\begin{array}{cc}{\omega }_1-i\frac{{\gamma }_1}{2}& g\\ g& {\omega }_2-i\frac{{\gamma }_2}{2}\end{array}\right)\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)\end{array},$$(1)where ${\omega }_{1,2}$ and ${\gamma }_{1,2}$ are the resonance frequencies and loss rates of the plasmon and exciton modes, and $g$ is the coupling coefficient. The complex energy eigenvalues of the coupled system are given by $$\omega ={\omega }_{\mathrm{ave}}-i{\gamma }_{\mathrm{ave}}\pm \sqrt{g^2+{\left({\omega }_{\mathrm{diff}}+i\frac{{\gamma }_{\mathrm{diff}}}{2}\right)}^2},$$(2)where ${\omega }_{\mathrm{ave}}=\frac{{\omega }_1+{\omega }_2}{2}$ and ${\gamma }_{\mathrm{ave}}=\frac{{\gamma }_1+{\gamma }_2}{2}$ are the average values of resonance frequencies and loss rates of the uncoupled plasmon and exciton modes, while ${\omega }_{\mathrm{diff}}=\frac{{\omega }_1-{\omega }_2}{2}$ and ${\gamma }_{\mathrm{diff}}=\frac{{\gamma }_1-{\gamma }_2}{2}$ are the differences between the resonance frequencies and loss rates. Generally, the coupling strength $g$ can be adjusted by the distance between the GNR and ${\mathrm{WSe}}_2$, while ${\omega }_{\mathrm{diff}}$ and ${\gamma }_{\mathrm{diff}}$ can be adjusted by the geometric size of the GNR. The complex eigenvalues of the hybrid system will coalesce when $\sqrt{g^2+{({\omega }_{\mathrm{diff}}+\mathit{i}{\gamma }_{\mathrm{diff}})}^2}=0$, and an EP can be achieved.

We use the scattering spectrum, which is experimentally accessible, to extract the resonance frequencies and loss rates of the hybrid modes. A typical scattering spectrum of the plasmon-exciton system is shown in Fig. 1(b). The scattering spectrum can be calculated by standard finite-difference time-domain (FDTD) simulation. By fitting the scattering spectrum with a double-Lorentzian function, we can obtain the resonance frequencies and loss rates of the hybrid modes, as shown in Fig. 1(b), which correspond to the real and imaginary parts of the eigenvalues of the system.

The coupling strength between the GNR and ${\mathrm{WSe}}_2$ is usually hard to precisely control in the experiment due to the nanoscale interaction length [59]. In order to overcome this difficulty, we propose to use the auxiliary layer of ${\mathrm{HfO}}_2$, the thickness of which can be accurately controlled with current experimental technology; e.g.,  atomic layer deposition equipment can be used to deposit a layer of ${\mathrm{HfO}}_2$ over ${\mathrm{WSe}}_2$ with nanometer resolution. Then, the GNR can be transferred onto the sample by spin coating in order to obtain the device depicted in Fig. 1(a), similar to the sample preparation in Ref. [59]. By adjusting the thickness of the auxiliary layer of ${\mathrm{HfO}}_2$, we can tune the coupling strength precisely between GNR and ${\mathrm{WSe}}_2$. Figure 1(c) shows typical scattering spectra for various thicknesses of the ${\mathrm{HfO}}_2$ layer. Another physical parameter we control is the intrinsic resonance frequency of the GNR. It is tunable by adjusting the length of the GNR. Figure 1(d) shows the scattering spectra for various lengths of GNR in the hybrid system.

The precise control over the coupling strength and relative resonance frequencies allows us to realize and observe the EP in the plasmon-exciton system. Figure 2(a) depicts the cross section view of the system. We tune the thickness of the ${\mathrm{HfO}}_2$ layer from 12 to 28 nm and the length of the GNR from 90 to 100 nm. For each choice of ${\mathrm{HfO}}_2$ thickness and GNR length, we numerically simulate the scattering spectrum of the system. Then, as described earlier, we fit each scattering spectrum to obtain the resonance frequencies and loss rates of the two hybrid modes, which correspond to the real and imaginary parts of the eigenspectrum of the hybrid system. Plotting the resonance frequencies and loss rates of the hybrid modes as a function of ${\mathrm{HfO}}_2$ thickness and GNR length, we obtain the eigenspectrum of the system. We can observe an EP at ${\mathrm{HfO}}_2$ thickness $\approx 20\mathrm{nm}$ and GNR length $\approx 93.6\mathrm{nm}$, where both the resonance frequencies and loss rates coalesce, as shown in Figs. 2(b) and 2(c). The eigenspectrum has the same topology as the Riemann surface of the complex function $f(z)=\sqrt{z}$. We also considered configuration II depicted in Fig. 2(d), where the GNR is placed directly on top of the glass substrate, parallel to the edge of the ${\mathrm{WSe}}_2$ monolayer. The distance between GNR and ${\mathrm{WSe}}_2$ tunes the coupling strength between plasmon and exciton modes. We plot the eigenspectrum of the hybrid system in the parameter space of GNR length and distance between GNR and ${\text{WSe}}_2$, and we observe an EP at a distance $\approx 14\mathrm{nm}$ with GNR length $\approx 107.6\mathrm{nm}$, as shown in Figs. 2(e) and 2(f). However, as this configuration requires more delicate experimental control techniques, we will use configuration I in Fig. 2(a) for the sensing application.

As we have observed the EP in the plasmon-exciton system, we continue to explore its application in EP-enhanced sensing. We first consider the sensing of environmental refractive index, which is essential for environmental monitoring and chemical sensing. Variation in the refractive index leads to the variation of the resonance frequency of the GNR, and this effect can be enhanced by the EP in the hybrid system. In order to simulate the sensing of the environmental refractive index, we add a cladding layer with a variable refractive index on top of the plasmon-exciton hybrid sensor. In Fig. 3, we compare the sensitivity of EP-enhanced sensing using a plasmon-exciton hybrid sensor (performed near the EP with ${\mathrm{HfO}}_2$ thickness $=20\mathrm{nm}$ and GNR length $=93.6\mathrm{nm}$) and regular sensing using a GNR-only sensor. In Figs. 3(a) and 3(b), we show the scattering spectra of EP-enhanced sensing and regular sensing for various refractive indices. In Fig. 3(c), we show the variation of resonance frequency difference [$\mathrm{\Delta }\omega =\mathrm{\Delta }({\omega }_1-{\omega }_2)$] and loss rate difference [$\mathrm{\Delta }\gamma =\mathrm{\Delta }({\gamma }_1-{\gamma }_2)$] between the hybrid modes of the plasmon-exciton sensor, as well as the variation of resonance frequency and loss rate of the GNR-only sensor, in response to the environmental refractive index change. Figure 3(d) shows the absolute value variation of the eigenvalue difference [$\mathrm{\Delta }E=\sqrt{{(\mathrm{\Delta }\omega )}^2+{(\mathrm{\Delta }\gamma /2)}^2}$] between the hybrid modes of the plasmon-exciton sensor and also the absolute value variation of the eigenvalue of the GNR-only sensor in response to the environmental refractive index change. As we can see from Fig. 3(d), the EP-enhanced plasmon-exciton sensor is more sensitive to environmental refractive index perturbation than the regular GNR-only sensor. The EP-enhanced sensing follows the square root signature near the EP with higher sensitivity, while regular sensing follows a linear trend with less sensitivity. The sensitivity factor of the regular GNR-only sensor is on the order of 100 THz per unit change of refractive index, which is consistent with earlier works [63,64], while the sensitivity factor of the EP-enhanced sensor can be more than 10 fold higher, depending on how close the sensor is to the EP. With an experimentally achievable resolution of 1 THz, the regular GNR-only sensor can detect a refractive index change on the order of 0.01, while the EP-enhanced sensor can detect a refractive index change on the order of 0.001. For comparison purposes, we also plot the sensing with a plasmon-exciton sensor whose parameters are set far away from the EP, and it shows a linear behavior similar to the regular GNR-only sensor without EP enhancement.

In addition to the EP-enhanced sensitivity, the plasmon-exciton hybrid sensor has the capability of nanoscale sensing for the environmental refractive index due to the sub-diffraction-limit size of the plasmonic resonator. We simulate the nanoscale sensing by a local refractive index variation within a box region ($200\mathrm{nm}\times 100\mathrm{nm}\times 100\mathrm{nm}$) surrounding the GNR, as shown in Fig. 4(a), instead of a homogenous refractive index variation, as discussed earlier in Fig. 3. We numerically calculate the absolute value variation of the eigenvalue difference [$\mathrm{\Delta }E=\sqrt{{(\mathrm{\Delta }\omega )}^2+{(\mathrm{\Delta }\gamma /2)}^2}$] between the hybrid modes of the plasmon-exciton sensor for both the nanoscale refractive index variation [green crosses in Fig. 4(b)] and the homogenous refractive index variation [red circles in Fig. 4(b) for comparison, the same as red circles in Fig. 3(d)]. We can see that the sensitivity of the EP-enhanced sensor for nanoscale refractive index variation agrees well with that for homogeneous environmental refractive index variation. This shows the nanoscale EP-enhanced sensing capability of the plasmon-exciton hybrid sensor.

The EP-enhanced sensor can also be used for nanoparticle sensing. For example, it can be used to check the length of a GNR in a non-invasive way. In this case, the GNR is both part of the plasmon-exciton hybrid system and the object to be measured. In Figs. 5(a) and 5(b), we show the scattering spectra of EP-enhanced sensing with plasmon-exciton hybrid modes and regular sensing with a single plasmonic mode for various GNR lengths. In Fig. 5(c), we show the variation of resonance frequency difference and loss rate difference between the hybrid modes and variation of resonance frequency and loss rate of the single plasmonic mode as the GNR length changes. In Fig. 5(d), we compare the absolute value variation of the eigenvalue difference between the hybrid modes and the absolute value variation of the eigenvalue of the single plasmonic mode. Again, we see that the EP-enhanced sensing with plasmon-exciton hybridization is more sensitive and follows the square root signature near the EP.

In conclusion, EPs are realized in plasmon-exciton hybrid systems. Plasmon-exciton sensors with EP-enhanced sensitivity can be used for nanoscale sensing of environmental refractive index changes and nanoparticles. They can also be used to detect other nanoparticles or materials that either change the effective refractive index around the GNR or modify the coupling between the plasmon and exciton modes and could find real-life applications such as environmental monitoring and biomolecule detection.